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L. (10 points) A biomedical researcher wants to test the effectiveness of a synthetic antitoxin. Twelve randomly selected subjects are tested for resistance to a particular poison. They are retested after receiving the antitoxin. Below are the differences in before and after readings for the subjects. (A positive value indicates that the subject's resistance to the poison has increased.) Use the Wilcoxon signed-rank test (with a= .05) to determine if the antitoxin tends to increase the subjects' resistance to the poison. Be sure to state the appropriate hypotheses and to give your conclusion in words.

• Make sure your exam has 6 different pages.

• You have two hours to complete the exam.

• You may use two formula sheets, a calculator and the statistical tables I posted at the website. (If a required degrees of freedom is not in the table, use the next smaller degrees of freedom that is in the table.)

• To get full credit show as much of your work as possible.

• Good luck and have a great summer!

Instructions:

Final Exam

Name STAT 212-502 May 10, 2010

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The data set included n. = 35 cases, the SSE for model 1 was 179.45047, and the SSE for model 2 was 183.58259. Use this information (with a= .05) to test the hypothesis that ,83 = 0 in model 1.

(b) (10 points) Another possible model (model 2) for the heights data is

Y = f3o + f31x1 + f32x2 + E.

(a) (8 points) Of interest is testing the hypothesis Ho : (33 = 0 in model 1. In words, what does this hypothesis imply about the male and female models?

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Y = f3o + f31.T1 + f32x2 + (33X1X2 + E, where Y is a student's height, x1 is 1 for female students and 0 for male students, and x2

is the height of a student's same-sex parent. This model implies that the females have the regression model Y = (30+(31 +(f32+,83)x2+E, while the males have model Y = f3o+f32x2+t.

2. Consider the following model (call it model 1) for some heights data that were collected in a previous 212 class:

2

(a) (8 points) Name the type of design used in this experiment and justify your answer.

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Subject Noise 1 2 3 4 5 6 7 Totals None 10 7 14 10 12 15 8 76 Pachelbel 10 8 13 8 11 12 9 71 Smashing Pumpkins 14 10 17 12 14 18 6 91

SST= 208.6667

3. A psychologist is studying the effect of background noise on reaction time. Each subject is exposed to three levels of background noise: no noise, classical music (Pachelbel's Canon), and rock music (Smashing Pumpkins, Siamese Dream). At each level, the subject has to perform a variety of tasks, and an overall estimate of his or her reaction time is determined. The particular ordering of the three noise levels is randomized for each subject. The results for seven randomly selected subjects follow.

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(c) (8 points) Using statistical methods we can supposedly make inferences about large populations based on relatively small samples. Would it be appropriate to draw a conclusion about the relationship, in general, between heights of children and their same-sex parents based on the data obtained from this STAT 212 class? Why or why not?

3

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(c) (10 points) Use Tukey's procedure with et= 0.05 to compare average reaction times for the three types of noise. State vour conclusions in words _,-- .L IM (} " · 11/c //'I. /rtea?

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(b) (10 points) Complete the following analysis of variance table.

Source of Degrees of Sum of Mean variation freedom squares square F Noise . ~ 30, 95 cl 38 /~Jl'f 6. 5.lf ~ Subject 0149.3333 ·----3> J"f,'l<f /tO.~J... Error /<A. J.?,3SIOJ.. ).3GS- Total J-0 so E'. 666 7

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where E is a random variable having the normal distribution with mean 0 and variance 14.

(a) (10 points) A park ranger is asked by his boss to give an educated guess for the number of Chaoborus larvae that would be found in a sample taken at a location where the lake is 10 meters Jeep and has a dissolved oxygen reading of 4 milligrams per liter. Help the ranger out by using the above equation to obtain an interval within which you're 95% surethelarvalcountis. FAc/ ),-5 a/le/ 97,!) /e__/'Ce/l/;/es.

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vfii ),5 jJe/'ce/JPk ;s /0. 0 7 (b) ( 10 points) At a certain spot in the lake the water depth is 5 meters and the dissolved

oxygen count is 2.5 milligrams per liter. What is the probability that the larval count will be no more than 11 for a sample taken at this point?

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Y = 15 + l.2x1 - 2.4x2 + E,

4. The phantom midge, genus Chaoborus, resembles the mosquito in appearance but not in bloodsucking habits. Let Y be the number of Chaoborus larvae collected in a grab sample of sediment from the bottom of a lake. Y is related to .T1, the lake depth (in meters) at the sampling point, and x2, the dissolved oxygen (milligrams per liter) in the water sampled from the lake bottom, in the following way:

5

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where e = the unknown probability that the coin comes up heads on any one toss. Would your methodology in answering (a) be any different in this case? Why or why not? (Note: You are not being asked to do any calculations; just describe how your analysis would differ, if any.)

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possibility that the coin isn't fair, the probability distribution of X (if the experiment were done correctly) would now be

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(a) (10 points) Does it appear that the students carried out the experiment in a conscien- tious fashion? Test the ap~ropriate hypothesis with ex = .01. ( ),)._

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Number of tosses The results of 175 students' experiments are summarized below.

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5. In the 211 class I taught in the 1992 fall semester I asked each of my students to conduct the following experiment. Toss a fair coin until heads appears, and then stop. Record the number of tosses that were performed. If X is one student's number of tosses, then X should have a geometric distribution with parameter .5. In other words, X should have the probability mass function

6